CLASS 9 NUMBER SYSTEMS POLYNOMIALS

 CLASS 9

NUMBER SYSTEMS

POLYNOMIALS

1. Find six rational numbers between 3 and 4.

2. Show how √5 can be represented on the number line.

3. Write the following in decimal form and say what kind of decimal expansion each has :

(i) 36/100

(II) 2/11

 

4. Given that 1/7 = 0.142857, can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how?

5. Express the following in the form p/q, where p and q are integers and q not equal to zero.

(i) 0.6666.....

(ii) $$0.4\overline{7}

6. Write three numbers whose decimal expansions are non-terminating non-recurring.

 

7. Find three different irrational numbers between the rational numbers 5/7 and 9/11.

8. Visualise 3.765 on the number line, using successive magnification.

9. Simplify:   (3+√3)(2+√2)

10. Represent (√9.3) on the number line.

 

11. Rationalize the denominators of the following:

(i) 1/(√5+√2)

(ii) 1/(√7-2)

12.  Find

(i)641/2

(ii) 322/5

$$0.4\overline{7} = 0.

(iii) 111/2/111/4

 (iv) 71/2×81/2

 

13. Find p(0), p(1) and p(2) for each of the following polynomials:

(i) p(y)=y2−y+1

(ii) p(t)=2+t+2t2−t3

14. Find the remainder when x3+3x2+3x+1 is divided by

(i) x+1

(ii) 5+2x

15. Check whether 7+3x is a factor of 3x3+7x.

16. Determine which of the following polynomials has (x + 1) a factor:

(i) x3+x2+x+1

(ii) x3 – x2– (2+√2)x +√2

17. Use the Factor Theorem to determine whether g(x) is a factor of p(x) given

 p(x)=x3+3x2+3x+1, g(x) = x+2

 

18. Find the value of k, if x–1 is a factor of p(x) in each of the following cases:

(i) p(x) = x2+x+k

(ii) p(x)=kx2–3x+k

 

19. Factorize:

(i) 12x2–7x+1

(ii) x3–2x2–x+2

(iii) x3–3x2–9x–5

(iv) 2y3+y2–2y–1

20. Use suitable identities to find the following products:

(i) (x+4)(x +10) 

(ii) (y2+3/2)(y2-3/2)

(iii) 104×96

(iv) 95×96  

21. Factorize the following using appropriate identities:

(i) 9x2+6xy+y2   

(ii)  x2–y2/100

22. Expand each of the following, using suitable identities:

 

(i) (−2x+3y+2z)2

(ii) (3a –7b–c)2

(iii) ((1/4)a-(1/2)b +1)2

23. Factorize:

(i) 4x2+9y2+16z2+12xy–24yz–16xz

(ii ) 2x2+y2+8z2–2√2xy+4√2yz–8xz

24. Write the following cubes in expanded form:

(i)    (x−(2/3)y)3

(ii)  ((3/2)x+1)3

   

25. Evaluate the following using suitable identities: 

(i) (99)3

(ii)  (998)3

26. Factorise each of the following:

 

(i) 64a3–27b3–144a2b+108ab2

(ii) 27p3–(1/216)−(9/2) p2+(1/4)p 

27. Factorize each of the following:

(i) 512y3 + 27z3

(ii)  125m3 – 343n3

28. Factorise: 27x3+ 64y3+ 125z3– 180xyz 

29. Without actually calculating the cubes, find the value of each of the following:

  (28)3+(−15)3+(−13)3

 30. Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: 

 Area : 25a2–35a+12

31. What are the possible expressions for the dimensions of the cuboids whose volumes are given below? 

 

  Volume : 12ky2+8ky–20k

32. If  x+y+z = 0, show that x3+y3+z3 = 3xyz.